Algorithms and basic asymptotics for generalized numerical semigroups in $${\mathbb {N}}^d$$ N d

Abstract

Let \({\mathbb {N}}\) denote the monoid of natural numbers. A numerical semigroup is a cofinite submonoid \(S\subseteq {\mathbb {N}}\). For the purposes of this paper, a generalized numerical semigroup (GNS) is a cofinite submonoid \(S\subseteq {\mathbb {N}}^d\). The cardinality of \({\mathbb {N}}^d \setminus S\) is called the genus. We describe a family of algorithms, parameterized by (relaxed) monomial orders, that can be used to generate trees of semigroups with each GNS appearing exactly once. Let \(N_{g,d}\) denote the number of generalized numerical semigroups \(S\subseteq {\mathbb {N}}^d\) of genus \(g\). We compute \(N_{g,d}\) for small values of \(g,d\) and provide coarse asymptotic bounds on \(N_{g,d}\) for large values of \(g,d\). For a fixed \(g\), we show that \(F_g(d)=N_{g,d}\) is a polynomial function of degree \(g\). We close with several open problems/conjectures related to the asymptotic growth of \(N_{g,d}\) and with suggestions for further avenues of research.